Jacobi-Trudi determinants over finite fields

Abstract

In this paper, we work toward answering the following question: given a uniformly random algebra homomorphism from the ring of symmetric functions over the integers to a finite field ${\mathbb{F}}_q$, what is the probability that the Schur function $s_{\lambda }$ maps to zero? We show that this probability is always at least $1/q$ and is asymptotically $1/q$. Moreover, we give a complete classification of all shapes that can achieve probability $1/q$. In addition, we identify certain families of shapes where the corresponding Schur functions being sent to zero are independent events, and we look into the probability that a Schur functions is mapped to nonzero values in ${\mathbb{F}}_q$.

39 pages